Peer-to-Peer Networks 03 CAN (Content Addressable Network) - PowerPoint PPT Presentation

Peer-to-Peer Networks 03 CAN (Content Addressable Network) Christian Ortolf Technical Faculty Computer-Networks and Telematics University of Freiburg

CAN Playground  Index entries are mapped to the square [0,1] 2 - using two hash functions to the real numbers - according to the search key  Assumption: - hash functions behave a like a random mapping 2

CAN Index Entries  Index entries are mapped to the square [0,1[^2 - using two hash functions to the real numbers - according to the search key  Assumption: Paul Francis Mark - hash functions behave a like Handley a random mapping  Literature Sylvia - Ratnasamy, S., Francis, P., Ratnasamy Handley, M., Karp, R., Shenker, S.: A scalable content- addressable network. In: Dick Computer Communication Karp Scott Review. Volume 31., Dept. of Shenker Elec. Eng. and Comp. Sci., University of California, Berkeley (2001) 161 – 172 3

First Peer in CAN  In the beginning there is one peer owning the whole square  All data is assigned to the (green) peer 4

CAN: The 2nd Peer Arrives  The new peer chooses a random point in the square - or uses a hash function applied to the peers Internet address  The peer looks up the owner of the point - and contacts the owner 5

CAN: 2nd Peer Has Settled Down  The new peer chooses a random point in the square - or uses a hash function applied to the peers Internet address  The peer looks up the owner of the point - and contacts the owner  The original owner divides his rectangle in the middle and shares the data with the new peer 6

3rd Peer 7

CAN: 3rd Peer 8

CAN: 4th Peer 9

CAN: 4th Peer Added 10

CAN: 5th Peer 11

CAN: All Peers Added 12

On the Size of a Peer‘s Area  R(p): rectangle of peer p  A(p): area of the R(p)  n: number of peers  area of playground square: 1  Lemma - For all peers we have  Lemma - Let P R,n denote the probability that no peers falls into an area R. Then we have 13

An Area Not being Hit  Lemma - Let P R,n denote the probability that no peers falls into an area R. Then R  Proof - Let x=Vol(R) - The probability that a peer does not fall into R is - The probability that n peers do not fall into R is - So, the probability is bounded by - because 14

How Fair Are the Data Balanced  Lemma - With probability n -c a rectangle of size (c ln n)/n is not further divided  Proof - Let P R,n denote the probability that no peers falls into an area R. Then we have  Every peer receives at most c (ln n) m/n elements - if all m elements are stored equally distributed over the area  While the average peer stores m/n elements  So, the number of data stored on a peer is bounded by c (ln n) times the average amount 15

Network Structure of CAN  Let d be the dimension of the square, cube, hyper-cube - 1: line - 2: square - 3: cube - 4: ...  Peers connect - if the areas of peers share a (d-1)-dimensional area - e.g. for d=2 if the rectangles touch by more than a point 16

Lookup in CAN  Compute the position of the index using the hash function on the key value  Forward lookup to the neighbored peer which is closer to the index  Expected number of hops for CAN in d dimensions: -  Average degree of a node - 17

Insertions in CAN = Random Tree  Random Tree - new leaves are inserted randomly - if node is internal then flip coin to forward to left or right sub-tree - if node is leaf then insert two leafs to this node  Depth of Tree - in the expectation: O(log n) - Depth O(log n) with high probability, i.e. 1-n -c  Observation - CAN inserts new peers like leafs in a random tree 18

Leaving Peers in CAN  If a peer leafs - he does not announce it  Neighbors continue testing on the neighborhood - to find out whether peer has left - the first neighboir who finds a missing neighbor takes over the area of the missing peer  Peers can be responsible for many rectangles  Repeated insertions and deletions of peers leed to fragmentation 19

Defragmentation — The Simple Case  To heal fragmented areas - from time to to time areas are freshly assigned  Every peer with at least two zones - erases smalles zone - finds replacement peer for this zone  1st case: neighboring zone is undivided - both peers are leafs in the random tree - transfer zone to the neighbor 20

Defragmentation — The Difficult Case  Every peer with at least two zones - erases smalles zone - finds replacement peer for this zone  2nd case: neighboring zone is further divided DFS - Perform DFS (depth first search) in neighbor tree until two neighbored leafs are found - Transfer the zone to one leaf which gives up his zone - Choose the other leaf to receive the latter zone 21

Improvements for CAN  More dimensions  Multiples realities  Distance metric for routing  Overloading of zones  Multiple hasing  Topology adapted network construction  Fairer partitioning  Caching, replication and hot-spot management 22

Higher Dimensions  Let d be the dimension of the square, cube, hyper- cube - 1: line - 2: square - 3: cube - 4: ...  The expected path length is O(n 1/d )  Average number of neighbors O(d) 23

More Realities  Build simultanously r CANs with the same peers  Each CAN is called a reality  For lookup - greedily jump between realities - choose reality with the closest distance to the target  Advantanges - robuster network - faster search 24

More Realities  Advantages - robuster - shorter paths 25

Realities vs. Dimensions  Dimensionens reduce the lookup path length more effciently  Realities produce more robust networks 26

Peer-to-Peer Networks 03 CAN (Content Addressable Network) Christian Ortolf Technical Faculty Computer-Networks and Telematics University of Freiburg